**Project Heads**

Martin Eigel, Martin Heida, Manuel Landstorfer

**Project Members**

Alireza Selahi

**Project Duration**

01.05.2021 − 30.04.2024

**Located at**

WIAS

Electrochemical electricity storage is a central pillar for a large variety of industrial goods, ranging from power sources for medical devices to electric vehicles and large scale battery plants. In 2019 this was honored by the Nobel price in chemistry, awarded “for the development of lithium-ion batteries (LIBs)”. The central innovation is the concept of intercalation, the physico-chemical process by which a lithium ion is stored within some solid host material. This process is essential for the safety, durability and energy density of modern LIBs.

However, all current and future LIBs face a common issue: they degrade in their lifetime upon usage. This degradation is in general a superposition of various ageing effects and depends on external (time dependent) parameters, e.g. the rate at which a battery is charged and discharged. Quantitative and qualitative knowledge of the degradation is of ultimate importance to estimate the lifespan of a battery, set up control engineering and ensure safety.

The project aims at developing a data-driven methodology to recover the dynamics of battery ageing on the basis of a parametrized mathematical model and experimental data. We want to determine the evolution of certain parameters of the model as function of the cycling number *N*. This is to be achieved by setting up a two time-scale PDE model, where the small time scale covers one charge/discharge cycle and the large time scale the number of such cycles.

To solve this statistical inverse problem for the degrading parameters, we will use recent ideas from invertible neural networks . Moreover, low-rank surrogate models for parametric PDE solutions will be employed in order to efficiently cope with the high model complexity [1,3].

In the long term, our approach can lead to real-time tracking and estimation of the battery health, which is of paramount importance for electric mobility. It can help to determine the residual value of an aged battery as well as its future lifetime, which is crucial for stationary energy storage devices and thus the “Energiewende”. The solution of statistical inverse problems with machine learning techniques is still in its infancy. We hope that we will contribute to the understanding of how to use Deep Neural Networks for these common problems, which will then be applicable in many fields.

Progress

Valid models of the physics in batteries are crucial in understanding the ageing mechanism and in developing new concepts to counteract the ageing. Landstorfer et al. [9] presented a modeling framework for simulations of parametrized LIBs based on homogenization in space, assuming only one active particle in a periodic unit cell, which is further assumed to be spherical.We extended the model to multiple active particles in one unit cell. A preprint is being written as of now.

Battery ageing is basically a two-timescale problem. Analogously to the space homogenization, we are deriving a homogenization in time. Currently, we are in the process of finding suitable scalings of smallness parameters and exploring the respective results. In particular, we consider two kinds of coupled problems: One is a diffusion problem, and the other one is a reaction problem. To tackle this problem, we are applying methods of asymptotic expansions to find possible homogenized problems and prove convergence using two-scale convergence theorems, among others.

du/dt = -∇ ⋅ ( D ∇u)

dD/dt = d(D, u, du/dt, ∇u)

Plus boundary/initial conditions

du/dt = -∇ ⋅ ( D(u,v) ∇u)

dv/dt = R(u,v)

Plus boundary/initial conditions

The degradation of a battery with each cycle is caused by many different phenomena. One cause of degradation is the cycle dependence of the diffusion coefficient D appearing in the PDE-model [9]. As a simplification, the time-dependent behavior is assumed to satisfy a simple evolution equation dD/dt = g(D(t)), which is an ODE in D with a right-hand side g.

Using a data-driven approach, we succesfully retrieved the rhs g based on charge-discharge-curves of batteries (which may be measurement data, or solutions of the forward problem in [9]). Since this is an inverse problem, we used Bayesian Inference. In particular, we applied Markov-Chain Monte-Carlo (MCMC) to sample from the posterior and additionally calculate the Maximum a-Posteriori (MAP) directly in cases where we are not interested in the uncertainty. A preprint is currently being written.

In battery ageing, multiple ageing effects can and do occur, for example a degradation of the reactivity as one effect, and a degradation of the diffusivity as another one. Depending on how the battery is used during its lifetime, different effects have a different impact on the ageing. Given the charge-discharge curves of a used battery, our goal is to identify which effect occurred with what impact. This way, we can make reliable predictions of the remaining lifetime of the battery.

The user behavior can be modeled as a curve, mapping the time time into an L-dimensional parameter space, where L is the number of occurring effects. This curve is parametrized with K weights, which represent the weights of the ageing effects. It is K ≥ L, since one effect progression can be modeled as a function of the time with more than one parameter.

The curve is then further mapped into an L-dimensional submanifold of an observation space X, which corresponds to the charge-discharge curves. Note that X may or may not be infinite dimensional. Given these observations, we successfully recovered the degradation weights of different ageing effects, both in a well-defined, invertible case, as well as in the case of an inverse problem. In the latter case, we again applied Bayesian Inference. A preprint is currently being written.

**Related Publications
**

- [1] M. Eigel, R. Schneider, P. Trunschke, and S. Wolf. Variational Monte Carlo – bridging concepts of machine learning and high dimensional pdes. Adv Comput Math, 2019.
- [2] M. Landstorfer. A discussion of the reaction rate and the cell voltage of an intercalation electrode during discharge. J. Electrochem. Soc., 167(1):A2573–A2589, 2019.
- [3] M. Eigel, M. Marschall, and R. Schneider. Sampling-free bayesian inversion with adaptive hierarchical tensor representations. Inverse Problems, 34(3):035010, 2018.
- [4] M. Heida and A. Mielke. Averaging of time-periodic dissipation potentials in rate- independent processes. Discrete & Continuous Dynamical Systems-Series S, 10(6), 2017.
- [5] M. Landstorfer. Boundary conditions for electrochemical interfaces. J. Electrochem. Soc., 164(11):E3671–E3685, 2017.
- [6] M. Heida. An extension of the stochastic two-scale convergence method and application. Asymptotic Analysis, 72(1-2):1–30, 2011.
- [7] M. Landstorfer, B. Prifling, and V. Schmidt. Mesh generation for periodic 3d microstructure models and computation of effective properties. Journal of Computational Physics, 431:110071, 2021.
- [8] M. Landstorfer. A Discussion of the Cell Voltage during Discharge of an Intercalation Electrode for Various C-Rates Based on Non-Equilibrium Thermodynamics and Numerical Simulations. Journal of The Electrochemical Society, 167(1):013518, November 2019.
- [9] M. Landstorfer et al., A Modeling Framework for Efficient Reduced Order Simulations of Parametrized Lithium-Ion Battery Cells, arXiv:2110.06011, 2021
- [10] J. Behrmann et al., Invertible Residual Networks, Proceedings of the 36th International Conference on Machine Learning, PMLR 97:573-582, 2019.

**Related Pictures
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